A NEW APPROACH TO ATTITUDE STABILITY AND CONTROLFOR LOW AIRSPEED VEHICLES

K.B. Lim∗, J-Y. Shin†, D.D. Moerder‡, E.G. Cooper§

NASA Langley Research CenterHampton, Virginia, USA 23681

Abstract

This paper describes an approach for controlling the at-titude of statically unsable thrust-levitated vehicles inhover or slow translation. The large thrust vector thatcharacterizes such vehicles can be modulated to providecontrol forces and moments to the airframe, but suchmodulation is accompanied by significant unsteady floweffects. These effects are difficult to model, and can com-promise the practical value of thrust vectoring in closed-loop attitude stability, even if the thrust vectoring ma-chinery has sufficient bandwidth for stabilization. Thestabilization approach described in this paper is based onusing internal angular momentum transfer devices for sta-bility, augmented by thrust vectoring for trim and other“outer loop” control functions. The three main compo-nents of this approach are: (1) a z-body axis angular mo-mentum bias enhances static attitude stability, reducingthe amount of control activity needed for stabilization,(2) optionally, gimbaled reaction wheels provide high-bandwidth control torques for additional stabilization, oragility, and (3) the resulting strongly coupled system dy-namics are controlled by a multivariable controller. Aflight test vehicle is described, and nonlinear simulationresults are provided that demonstrate the efficacy of theapproach.

1 Introduction

1.1 Background

There has been recent interest in flight vehicles, capableof VTOL and hover, that are statically unstable or neu-trally stable [1]. Examples of such vehicles include smallobservation platforms [2], [3] in which an optical sensoris mounted above a levitation-producing ducted fan, andrescue vehicles [4] consisting of a payload-bearing plat-

∗Senior Research Engineer, Guidance and Controls Branch,kyong.b.lim@nasa.gov

†Staff Scientist, National Institute of Aerospace,j.y.shin@larc.nasa.gov

‡Senior Research Engineer, Guidance and Controls Branch,d.d.moerder@larc.nasa.gov

§Senior Research Engineer, Systems Integration Branch,e.g.cooper@larc.nasa.gov

form with a number of ducted fans distributed symmetri-cally about its perimeter in the plane of the platform. Inthe state of the art, this type of vehicle is controlled bythrust vector variation, possibly with additional aerody-namic features implemented to enhance stability in thepresence of gusts [2], [3].

Using thrust vectoring or differential thrusting to sta-bilize a statically unstable airframe poses two related dif-ficulties. First, because of the airframe’s lack of stabil-ity, the controller must operate with high authority andbandwidth in order to contain and suppress excursionsin attitude. Accurate system models are generally re-quired to successfully design such controllers. The seconddifficulty is that such models are not available. This isbecause the active thrust vector variations required forstabilization in the presence of disturbances induces un-steady flow phenomena that distort the control commandin a complicated manner that currently defies compre-hensive modelling (see for example [5]. These difficul-ties come together when the control engineer attempts todesign the required high-performance controller while ig-noring or under-modelling the effect of the unsteady flowphenomena on the closed-loop dynamics These difficul-ties become more evident for smaller inertia VTOL vehi-cles because they respond more rapidly to disturbancesso that correspondingly more rapid feedback control isnecessary for vehicle stabilization.

1.2 A New Approach

This paper describes the configuration, dynamics, andcontrol of a statically unstable flight vehicle that levi-tates by means of thrusters with thrust vectoring devices,but augments the thrust vector control with internal an-gular momentum transfers to enhance attitude stabilityand provide secure command tracking performance in thepresence of external disturbances, such as wind gusts andin-flight payload variations.

The salient features of such a vehicle are:

• The vertical component of the thrust vector levitatesthe vehicle.

• The thrust vector is produced by a plurality of thrustgenerators, such as ducted fans. The direction of thethrust vector is modified by changing the orientation

https://ntrs.nasa.gov/search.jsp?R=20040095917 2018-07-17T02:09:23+00:00Z

of the thrust generators, such as by gimballing, or bydiverting their flow. The contributions from the plu-rality of thrusters produce net forces and momentsabout the airframe’s center of mass that can be usedto trim the vehicle and for control. For the exam-ple system explored further in the sequel, thrust isprovided by four ducted fans, each with axis of rota-tion fixed parallel to the z body axis, and the thrustdirection is modulated by a pair of thrust vectoringvanes under each fan.

• Static attitude stability is enhanced by an angularmomentum bias in the z body axis. This bias is pro-duced by a plurality of rotating masses, such as ded-icated momentum wheels. Alternatively, when em-ploying fans or impellers whose rotation is alignedwith the z body axis for levitation thrust, the z-body angular momentum bias can be generated byaugmenting their moments of inertia about their ro-tation axes. This is done by redistributing mass to-ward, or introducing additional mass near the fan orimpeller radius for one or more of the fans or im-pellers. An example of this is attaching a ring to thetips of a levitation fan.

• Additional attitude control authority is provided bya plurality of control moment gyros (CMGs). Theseinteract with the thrust vector control in jointlygenerating moments for attitude stabilization; thethrust vectoring system is also used to desaturatethe CMGs.

• The control effectors described above are jointlycommanded by a control system for stabilizationand command tracking, and for desaturation of theCMGs.

The advantages of the above vehicle are:

• The vehicle’s open-loop attitude stability is enhancedby the z-body axis angular momentum bias. Thisreduces the authority required from the active stabi-lization control system, and permits a reduction inthe bandwidth of the thrust vectoring control com-mands.

• It is straightforward to develop accurate control de-sign models of the devices for internal angular mo-mentum exchanges. Because of this, they can beused in high-bandwidth, high-authority control sys-tems. This permits enhanced disturbance rejectionsand maneuverability.

• The use of internal angular momentum exchange de-vices relaxes the performance requirements on thethrust vector control effectors, since their role can bereduced to as little as that of providing trim, com-mand tracking, and CMG desaturation.

Figure 1 illustrates the basic differences between thestate-of-the-art approach and our new approach.

Figure 1: Inner-loop attitude control by differential andthrust vectoring, state-of-the-art (upper), and by internaltorques aided by bias momentum, our approach (lower).

1.3 Outline of Paper

This paper outlines recent results in our continuing R&Defforts as first reported in [6]. In the next section, a re-search vehicle under development at NASA Langley Re-search Center for maturing this control technology is de-scribed, and its equations of motion are given. Section 3discuss its flight dynamics and control aspects includingvehicle trim, gyric stability, and control structure. Thesizing of the bias momentum for this new class of “hover-ing spacecraft” vehicles is also discussed. In section 4, wedescribe a current flying test platform system under de-velopment and outline its simulation-based performanceassessment and comparison with a similar vehicle not us-ing momentum exchange devices. Section 5 provides con-clusions.

2 Equations of Motion

To derive the equations of motion for a generic NFTPsystem1, a Newton-Euler formulation is used, and in par-ticular, the sequence of formulation and notation foundin [7], and extended to multiple gimbaled spinning bodieswith vane rotational motion for vector thrusting. Only asummary of the results are given here and the interestedreader is referred to [8] for details.

Figure 2 shows a schematic of the NFTP describingits main components.

1“Generic” in the sense of a flying platform propelled by multipleducted-fan and vane system and augmented with multiple wheelsystem

2

Figure 2: Schematic of NFTP.

R Rigid platformFi i-th Fan: Propeller(Pi) + Motor(Mi)Ci i-th CMG: Wheel(Wi) + Gimbal(Gi)B Momentum wheel, centered at Ob

Ob origin of platform fixed axesOc c/m of system (S = R+

∑i(Fi + Ci) +B)

Ocbc/m of R

OFiorigin of i-th ducted fan subsystem axes

OGi origin of i-th CMG frameLet ωsi

ai, ΩB z, and ηigi, denote the relative angularvelocities of the ith fan, momentum wheel, and the gimbalsystem relative to platform, respectively. Additionally,let Ωisi), denote the relative angular velocity of the ithCMG wheel relative to its gimbal frame. For the CMG,assume that the center of mass of rotor is fixed in Fb,and that the gimbal axis, g, is fixed in Fb, while the spinand output axes are defined as s and o respectively. Formore details on CMGs, see for example [9]. We define thefollowing frames:

Fo , bases for inertial frame, origin OFb , (x, y, z)T = CFo bases for platform frameFFi , CFiFb bases for i-th ducted fan frameF o

Gi, Co

GiFb = (so

i , gi, ooi )

T initial i-th gimbal frame

FGi , (si, gi, oi)T i-th gimbal frameFGi , CGi(ηi)F o

Gi= CGiFb

CGi(ηi) ,

cos ηi 0 − sin ηi

0 1 0sin ηi 0 cos ηi

In summary, we consider an 19 degrees of freedom NFTPmodel which includes the following:

3 positions of Ob relative to O3 rotations of R relative to Fo

4 rotations of fan relative to Fb

1 rotation of B relative to Fb

4 rotations of CMG gimbals relative to Fb

4 rotations of CMG wheels relative to FGi

The following momenta variables are used in describ-ing the equations of motion:

p−→

= m v−→+ ω−→× c−→ (1)

h−→ = c−→× v−→+ J−→ · ω−→+4∑

i=1

IaγFi

ωsiai

+4∑

i=1

(IgCiηigi + Ia

WiΩisi) + Ia

BΩB z (2)

hai = IaFiai · ω−→+ Ia

γFiωsi , i = 1, . . . , 4 (3)

hBz

= IaB z · (ω−→+ ΩB z) (4)

hWi1 = IaWi

(ωηi

1 + Ωi), i = 1, . . . , 4 (5)hCi2 = Ig

Ci(ωηi

2 + ηi), i = 1, . . . , 4 (6)

The vectors, v−→ and ω−→ denote the platform translationaland rotational inertial velocities, respectively, while c−→denotes the first moment of inertia about Ob for the sys-tem having a mass m. The vectors, p

−→and h−→, denote the

the translational and rotational momentum of the sys-tem, respectively. The angular velocity term, ωηi

j wherei = 1, . . . , 4 and j = 1, 2, 3 (corresponding to si, gi, andoi), denotes the platform angular velocity components inthe jth axis of the ithe gimbal frame. In particular, if werepresent the platform angular velocity in terms of bodyframe so that ω−→ = FT

b ω, the above angular velocity termscan be expressed in matrix form as

ωη1 = Kω, ωη

2 = Koω, ωη3 = Gω (7)

where

K ,

k(η1)k(η2)k(η3)k(η4)

, Ko ,

ko1

ko2

ko3

ko4

, G ,

g(η1)g(η2)g(η3)g(η4)

and for i = 1, . . . , 4

k(ηi) ,[

cos ηi 0 − sin ηi

]Co

Gi(8)

koi,

[0 1 0

]Co

Gi(9)

g(ηi) ,[

sin ηi 0 cos ηi

]Co

Gi(10)

For practical reasons, the rotational momentum isdefined about the point Ob, which is the center of theplatform frame, Fb, but need not be the center of massof S due to practical limitations and/or not accuratelyknown variable mass payload and its distribution duringflight. Naturally additional complications arise due to thefact that reference point Ob is neither the center of massnor fixed to an inertial frame. The remaining variables,hai

, hBz, hWi1 , hCi2 , are absolute rotational momenta vec-

tor components of the ith rotating fan, momentum wheel,CMG wheel, and CMG system, along their respective pri-mary spin axes (i.e. ai,z,si,gi). As reference points forthe above momenta components, their respective centerof masses are used. Of course the center of masses for

3

these actuator subsystem hardware can be accurately de-termined and are not expected to vary during operation.

With the above choice of variables and notations, theequations of motion are summarized as follows (for detailssee [8]):

p−→

= m g−→

+

4∑i=1

T−→i+ f−→jet

+ f−→aero

(11)

h−→ = − v−→× p−→

+ c−→× g−→

+

4∑i=1

[ τ−→i+ b−→Fi

× T−→i]

+ τ−→jet+ τ−→aero

(12)

hai = τFi(13)

hBz= τBz

(14)

hWi1 = τsi (15)

hCi2 = −ωηi3

[(Is

Ci− Io

Ci)ωηi

1 + IaWi

Ωi

]+ τgi (16)

where i = 1, . . . , 4, i.e., 4 sets of ducted fans and sin-gle gimbal variable speed CMGs are assumed, but clearlyany number can be used in these equations. The terms,T−→i

, T−→Wi

+T−→bi

, τ−→i, τ−→Wi

+ τ−→bi

, denote external aero-propulsive net thrust and torques on the system due tovector thrusting and differential throttling of the ductedfan and vane subsystem. The terms f

−→jet, f−→aero

, τ−→jet,

and τ−→aerodenotes the aerodynamic loads on the system

due to significant vehicle airspeed. In this paper, we focuson the subclass of problems where the vehicle is flying atlow airspeeds, due to its particular difficulty in attitudestabilization. Hence we do not include these terms in thispaper but is addressed in [8]. The terms τsi and τgi de-notes the internal torques on the i-th CMG wheel due tothe wheel motor, the internal torques on the i-th CMGsubsystem due to the gimbal motor, respectively. Thevariable τ

Bzdenotes the internal torque on the momen-

tum wheel due to its motor, while τFi

denotes net torqueon the i-th fan due to a combination of external fan rota-tional drag and internal fan motor torque, including backEMF effects.

2.1 Aerodynamic and propulsive loads

Figure 3 shows a set of frames used to conveniently de-scribe various loads produced by engine thrust, vane de-flections, and fan rotational drag.

2.1.1 Net propulsive forces acting on OFi

T−→i(ωsi

, θi) = FTb C

TFiTi(ωsi

, θi), Ti ∈ R3×1 (17)

where

Ti(ωsi , θi) ≈

cTxi

cTyi

cTzi

ω2si

+

cxiθi

cyiθi

−cziθ2i

ω2si

:= ei(θi)ω2si

(18)

bF

oF

Ti

OG i

Ob

gi

^

FG i

s i^

biC

F ib

F iO

ai

FFi z

y

x

Z

Y

X

O

τ i

Figure 3: Reference frames for modeling loads producedby Fans, Vanes, and CMG.

2.1.2 Net propulsive moments acting about OFi

τ−→i(ωsi

, θi) = FTb C

TFiτi(ωsi

, θi), τi ∈ R3×1 (19)

where

τi(ωsi , θi) ≈

cτxi

cτyi

cτzi

ω2si

+

lxi

lyi

lzi

θiω2si

:= fi(θi)ω2si

(20)

2.1.3 Rotational Drag on i-th Fan along ai

ai · τ−→Wi

= −τWi

(ωsi), τWi∈ R (21)

whereτWi

(ωsi) ≈ cRiω2

si(22)

Fan rotational drag only in ai axis (i.e. excludes vanedeflection effects).

2.1.4 Torques on i-th Fan motor along ai

ai ·M−→Wi

, τai(ωsi

, ωcmdsi

), τai∈ R, i = 1, . . . , 4 (23)

whereτai(ωsi , ω

cmdsi

) ≈ kEMF iωsi + k3ω

cmdsi

(24)

The torque modeled in equation 24 is a combination ofback EMF and armature current torques. The net torqueon the fan system along spin axis due to rotational dragand motor

τFi

, −τWi

(ωsi) + τai(ωsi , ωcmdsi

) (25)

4

2.2 Scalar form

For modeling and computational purposes it is more con-venient to express the motion equations in scalar/matrixform. In particular, we choose to express vector quan-tities in terms of their components in Fb, FFi

, and FGi

frames:

ai = FTb ai

b−→Fi

= FTb bFi

c−→ = FTb c

J−→ = FTb JFb

h−→Gi

= FTGihGi

ω−→Gi

= FTGiωGi

g−→

= FTb Cgo

p−→

= FTb p

h−→ = FTb h

ω−→ = FTb ω

v−→ = FTb v

T−→i= FT

FiTi

τ−→i= FT

Fiτi

where all column matrix variables belong to R3×1 exceptJ ∈ R3×3. These quantities include geometrical and iner-tial parameters of the vehicle that are constants in bodyframe, and state variables and forces and moments whosecomponents can vary in body, gimbal, or fan frames.

The first order momenta equations for the total sys-tem are summarized as follows

p

h

=

[−ω× 0−v× −ω×

]ph

+

[mI3×3

c×

]Cgo

+[CT

F 0CbF CT

F

]T (ωs, θ)τ(ωs, θ)

(26)

and for ducted fan along spin axis are

ha = τF(ωs, ω

cmds ) (27)

and for momentum wheel along spin axis, z

hBz

= τBz

(ΩB ,ΩcmdB ) (28)

and for CMGs along spin and gimbal axes arehW·1

hC·2

=

τs

−diag(Gω) [(IsC − Io

C)Kω + IaW Ω] + τg

(29)

The momenta is related to the velocities as follows

phha

hBz

hW·1

hC·2

=

[M Φ(η)T

Φ(η) Ψ

]

vωωs

ΩB

Ωη

(30)

where M ,

[mI3×3 −c×c× J

], Φ(η) ,

04×3 IaFA

T

01×3 (0 0 IaB)

04×3 IaWK

04×3 IgCKo

, and Ψ , diag(IaγF , I

aB , I

aW , Ig

C

).

In the above choice of dynamical equation coordinates,the inertial attitude of the platform (i.e. C) enters

through the effects of gravity. Its inadvertent effects onrotation is due to the limited knowledge in the exactlocation of the system center of gravity. If the centerof gravity of the whole system is known, then, Ob canbe chosen as such and the rotational and translationalmotions uncouple.

In the above equations, the superscript (·)× denotesthe skew-symmetric matrix associated with vector cross

products, for example, ω× ,

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

.

For the definition of augmented states and variables in-cluding ωs, ha,Ω, η, hW·1 , hC·2 , T, τ, τW

, τa, τs, τg, τF

,and augmented parameters includingCF , CbF , A, I

aM , Ia

P , IaF , I

aγF , I

aW , Ig

C , see [8] for details.For kinematics, we denote the position of the ori-

gin, Ob, of the platform-fixed frame, Fb, with respectto inertial frame, Fo as R−→Ob

= FTo ξ so that the iner-

tial velocity of Ob is v−→ = FTb v = FT

o ξ, and the iner-tial velocity components in Fo frame can be written asξ = C(β)T v where we choose to parameterize the direc-tion cosine matrix relating Fo to Fb by Euler Parameters,β := (β0, β1, β2, β3)T ∈ R4×1. For the form of this direc-tion cosine matrix parameterization and the time deriva-tive of these Euler parameters, see for example [10].

3 Flight Dynamics & Control

3.1 Trim

We initially consider the flight dynamics as referencedto trim conditions which we define as attitude hold whilemoving at constant translational speeds. This of course issuited only for cruise and hovering flight segments. Thetrim conditions are found by calculating any combina-tions of four vane angles and four fan speeds necessary tohold a given desired attitude, at a constant translationalvelocity. With variable speed CMGs we choose a nom-inal wheel speeds and zero gimbal rotation rate as thetrim value. In summary, the trim conditions are definedby setting all momenta state derivatives, given in equa-tions 26 to 29, to zero. In addition, we consider ω = 03×1

to hold platform attitude to β.Assuming the above conditions, the first order mo-

menta equations for trim reduces to a coupled set of alge-braic equations [8]. We make the following observationsfrom the trim conditions:

• Because ω = 0, p = mv, i.e. at trim, the linearmomenta do not affect rotational equilibrium, i.e.,v×p = 0.

• For a given steady or near trim flight condition, acombination of vector thrusting via fan speeds, ωs,and vane angle deflections, θ, are used to satisfy trimconditions. In the case when the vehicle is requiredto trim at airspeeds that result in significant vehicle

5

aerodynamic loads, as described in equations 11 and12, the above trim conditions can be slightly modi-fied.

• For the ducted fan trim ˙ha = 0, the fan control mo-

ments, τa, exactly balance the fan rotational drag,τ

Was given in equation (25). The trim command

can be computed once the required fan speed ωs isknown from system momenta equilibrium equations.

• For the momentum wheel trim, hBz= 0, the wheel

control moments, τmotorB

exactly balances the wheelfriction, τfriction

B. The desired level of bias momen-

tum for the overall system defines the momentumwheel trim speed,ΩB , which will determine its corre-sponding command, Ωcmd

B .

• For each CMG wheel trim where ˆhW·1 = 0, the CMG

wheel motor must exactly balance the CMG wheelfriction at a chosen trim speed Ω.

• For each CMG gimbal system trim where ˆhC·2 = 0,

the CMG gimbal motor torque must be zero corre-sponding to gimbal rate trim, ˆη = 0.

The trim conditions can be expressed only in terms of thetrim angles, θ, and fan angular velocities, ωs, by using thevector thrusting models as given in equations (17) to (24)for the forces and moments produced by fan rotation andvane deflections. These key subset of trim conditions canbe rearraged to the following compact form:

M(θ)ω2s + φ(β) = 06×1 (31)

where

M(θ) :=[

CTF

1boCbF

]E(θ) +

[0

1boCT

F

]F (θ) (32)

φ(β) :=[mI3×3

1boc×

]C(β)go (33)

define the geometry, kinematics, mass properties, sec-ondary trim actuators, and vehicle aerodynamics. Thescalar constant bo meters is a reference moment arm tohelp scale the combined force-moment equations to forceunits, which helps in the selection of a treshold numericalvalue for defining equilibrium.

For any given set of variables, (β, θ, ωs, v), that satis-fies the vector thrusting trim equations (31), one can de-duce the corresponding ducted fan speed trim commandωcmd

si. The remaining trim commands, momentum wheel

speed ΩcmdB , CMG wheel speed Ωcmd, and CMG gim-

bal rate ˆηcmd, will clearly depend on their correspondingdesired trim conditions (ΩB , Ω, ˆη = 0), and their corre-sponding motor dynamics. Note that these details involvemotor commands to regulate and track armature rotationspeeds, and involve technology that is well understoodand developed.

Proposition 1 (Trimability) Given a platform atti-tude, β, the system defined by equations (26)-(29) istrimmable, using vector thrusting (using a combination ofcontrol vane deflections, θ, fan thrusts with fan rotationspeeds ωs, and reaction jets), if and only if there exists θsuch that φ(β) ∈ Range[M(θ)].

If the above trim conditions are met for β and θ, then,the corresponding set of all possible fan speed combina-tions can be parameterized as follows:

ω2s = −M(θ)+φ(β) + ψ where ψ ∈ Null[M(θ)] (34)

Proposition 2 (Numerical Criteria for Trim)Given a platform attitude, β, the system defined byequations (26)-(29) is trimmable, using vector thrustingif the following least squares equation error is satisfied:

‖ε(β, θ)‖2 := ‖[I −M(θ)M(θ)+]φ(β)‖2 ≤ ε (35)

The superscript symbol in (·)+ denotes its pseudo-inverse.Note that the treshold for zero equation error denoted as εshould reflect practical limitations including (i) numericalerror, in the order of machine epsilon, (ii) accuracy limi-tation in the actuation and measurement of vane angles,(iii)accuracy limitation in the fan thrust response to com-mand, including integer command resolution, and (iv)sensor accuracy limitaion in a high quality accelerometer.These factors are used to define an engineering tresholdof “trim” for the NFTP during low airspeed operations.Since trim angles and speeds are non-unique if they ex-ists, we so choose the “smoothest” combination, see [8].

3.2 Gyric Stability about trim

Figure 4 shows the perturbed dynamics about trim. In

Figure 4: Perturbed dynamics about trim.

general, the open loop dynamics involves complex per-turbed response due to gyroscopic coupling in the rota-tion axes and the coupling with translational degrees offreedom (in addition to coning due to axis misalignmentsand translational-rotational coupling due to off CG ef-fects). Hence for the sake of analytical tractability, wesimplify by assuming a perfect knowledge of system CGlocation and the orientation of the system principal axes.With these assumptions, the tenth-order system charac-teristic equation for the linearly perturbed system shown

6

in Figure 4 can be written and solved explicitly. It turnsout that the only non-zero eigenvalues have the form

λ9,10 = ±jλo, where λo :=|hB |√J1J2

(36)

Physically, this complex conjugate pair corresponds tothe gyroscopic coupling in the pitch and roll axes. Thisoscillation or wobbling frequency given in equation 36 isanalogous to the “precession frequency” in a dual-spinspacecraft (a gyrosat with a nonspinning carrier) as de-rived for example in Chapter 6 of [7]. Clearly, this wob-bling frequency increases linearly with the level of biasmomentum and is inversely proportional to the rotationalinertia of the vehicle. It can be shown that the transferfunction matrix from external torques (ν1, ν2, ν3) to smallangular responses in roll, pitch, and yaw has the form

δβ1

δβ2

δβ3

=

[ 1sG(s; hB , J1, J2) 02×1

01×21

s2J3

] ν1

ν2

ν3

(37)

where the roll-pitch transfer matrix is

G(s;hB , J1, J2) ,1

s2 + λ2o

[sJ1

− hB

J1J2hB

J1J2

sJ2

](38)

Figure 5 shows the roll angle frequency response from dis-turbance torques in the roll and pitch axes. Notice thatthe linear yaw response depends only on the yaw distur-bance torque and is independent of roll and pitch motion.The main point is that bias momentum can significantlyimprove attitude stability by reducing the response to dis-turbances in roll and pitch. Furthermore, as the level ofbias momentum increases, the roll response to roll distur-bances decreases more rapidly than to pitch disturbancetorque. Alternately, the dynamically uncoupled systemat zero bias momentum transforms to a system whosecharacteristic dynamics is gyrically dominated.

Figure 5: Open loop frequency response in roll and pitchangles at different levels of bias momentum.

3.3 Bias Momentum Sizing

Consider the rotational dynamics of a generic dual-spinsystem with principal rotational inertias (J1,J2,J2) drivenby disturbance torques (τ1,τ2, and τ3) in all three axes.Suppose the main body rotates with angular velocity

components in body coordinates denoted by (ω1,ω2, andω3) while the momentum wheel spins about the verti-cal (3-axis) at a fixed rate relative to the main body toproduce a constant bias momentum of hB . In addition,suppose the bias momentum in the wheel “dominates” inthe following sense

|hB | max (|(J3 − J2)ω3|, |(J3 − J1)ω3|) (39)

Then, the dynamics of this generic dual spin system canbe approximated by

J1ω1 ≈ −ω2hB + τ1 (40)J2ω2 ≈ ω1hB + τ2 (41)J3ω3 = (J2 − J1)ω1ω2 + τ3 (42)

These equations are not limited to “small” angular mo-tion (associated with linearizations) but rather to thedominance of the bias momentum, as defined in equation39. Although the yaw dynamics in equation 42 is nonlin-ear, the roll-pitch dynamics in equations 40 and 41 arelinear. In fact this transfer function matrix for roll-pitchdynamics is identical (except 1/s term) to the dual-spinlinearized roll-pitch dynamical equations 38.

For this generic dual spin system, we consider a biasmomentum sizing approach based on the angular velocityresponse of the main body to a set of disturbances. Toobtain an analytical expression for roll-pitch response torandom disturbance torques, we make some mild assump-tions on these unknown disturbances. Suppose the powerspectral densities of the disturbance torques caused byunknown random turbulence, τ1, and τ2, with a commonbandwidth of BWτ , can be modeled as

V (jν) , diag(ρτ1 , ρτ2), ∀ν < BWτ (43)

where ν denotes the dummy frequency parameter. It canbe shown that [8] the mean square response of the rolland pitch angular velocities can be expressed as follows

E [ω21 + ω2

2 ] ≈√

2 max(ρτ1 , ρτ2)√J1J2|hB |

(44)

where the above estimate applies for responses at lowerfrequencies ν < min(BWτ , λo). Figure 6 shows the effectof scaling bias momentum on the worst case angular ve-locity response in the roll and pitch axis, in terms of themaximum singular value frequency response. It followsthen that if we want the RMS angular velocity responseto be bounded by a certain maximum desirable value,then the bias momentum can be sized to satisfy an in-equality based on equation 44. In addition to the aboverequirement on the RMS response of the angular veloc-ities, the sizing of the bias momentum should be suchthat the precession frequency lies outside the bandwidthof the disturbance, BWτ . This bandwidth requirementcan be satisfied if we choose bias momentum level to sat-isfy, based on equation 36, the following condition:

λo :=|hB |√J1J2

> BWτ (45)

7

Figure 6: Maximum singular value frequency response inroll and pitch anglular velocities for sizing bias momenta.

3.4 Control system framework

Figure 7 shows a flight dynamics and control schematicfor a general NFTP vehicle with the use of both biasmomentum wheel and a set of variable speed CMG’s.The vector thrusting commands include four fan speeds

Figure 7: Flight dynamics and control schematic.

(ωcmds ) and four vane angles (θcmd) to control platform at-

titude (β), and velocities (v and ω). These vector thrust-ing commands for the fans and vane angles are primarilyused for generating trim conditions and for open loopcommand tracking. For the general configuration withCMGs, a secondary but nevertheless crucial function ofthe vector thrusting commands will be to stabilize theCMGs so that their saturation tendencies will be miti-gated. To improve performance, fan speed regulators andvane servos are used.

Notice that the Ducted Fan/Vane aerodynamics andgravity directly influence system momenta p, h, while theCMG Dynamics has no influence on system momenta.

Rather, the CMGs internally redistribute angular mo-mentum to control the platform angular velocity ω and itsattitude β. A key advantage is that the CMG can gener-ate internal control torques to effectively and very reliablyredistribute angular momentum to control the platform,independently of Ducted Fan/Vane aerodynamics. Thisof course mitigates the uncertain but significant effectsof unsteady aerodynamics induced by rapid control sur-face motions for attitude stabilization. This property ofthis control approach will particularly benefit smaller ve-hicles during hovering or operations at low airspeeds un-der significant wind turbulence because they will respondrapidly due to their smaller rotational inertias and willtherefore require control effectors that can reliably gener-ate control forces and moments at higher bandwidth forattitude stabilization.

While the bias momentum wheel is regulated to a cer-tain constant speed, ΩB , to provide directional stabilityin open loop, the CMG subsystem will consist of closedloop torque commands to gimbal and wheel motors forvariable speed CMGs. These commands will be generatedby a control law which will integrate the CMGs with thevector thrusting control system for a given momentumwheel augmented platform. In particular, an approachwith extends the steering laws outlined in [16], [17], and[15] to account for factors including, trim requirements,gravity effects, significant persistent air turbulence, trans-lational coupling dynamics, and CMG desaturation usingducted fan based vector thrusting has been developed.

An alternate control viewpoint is based on the lineardynamics in Figure 4. This perturbed dynamical systemcan be viewed as an linear parameter varying (LPV) sys-tem for control law design wherein the scheduled param-eters may include the trim parameters such as attitudeand translational velocity. This viewpoint is particularlyattractive because the control law analysis and synthe-sis is well understood and technologically viable (see forexample [11],[12],[13],[14]). For further details on bothcontrol law approaches, see [8].

4 Flying Test Platform

4.1 System overview

The attitude control approach described above is testedusing the vehicle pictured in Figure 8 as a CAD model.The vehicle is a platform levitated by four ducted fansarranged symmetrically about the z-body axis. Each fanduct is equipped with a parallel pair of thrust vectoringvanes, downstream from its fan. Each fan and pair ofvanes is individually commanded by the control system.The z-body axis angular momentum bias is provided by amomentum wheel, powered separately from the levitationfans, and can be seen in the Figure at the center of theplatform. Although the simplest and lightest momentumbiasing mechanism is provided by augmenting the z-body

8

axis moment of inertia of the propulsion system, momen-tum wheel is separately mechnized in order to facilitatetests in which different magnitudes of momentum biasare explored. For initial testing, the vehicle will operatewithout CMGs or reaction wheels.

In its initial configuration, the fan motors andmomentum wheel motor are Aveox F27-F5B, and JRDS8411 are the vane servos which provides 300 deg/smaximum deflection rate. Each of the four levitationfan motors draw two kilowatts at full thrust. Becauseof this, the vehicle will be fed power from an overheadtether whose tension is controlled to offset its weight. Thetether will mount to the platform at the vehicle’s centerof mass, to minimize tether-induced moment disturbancesduring flight. The dynamics of the tether are ignored inthe next Section’s simulation model. The attitude sta-bilization software is hosted on a PC104 computer thatis affixed to the platform, and which communicates witha ground-based dSpace real-time control system. Thislatter is used to generate trim and guidance commands.The platform hosts an IMU model MIDG II, which issupplemented by a machine vision subsystem that givesEarth-fixed position and attitude. More details of thissystem are given in [8].

Each fan produces roughly 10 lbf of thrust at fullpower, which is more than adequate to levitate the grossplatform mass of approximately 29 lbm. The moments ofinertia (in Kg-M2 units) for this vehicle are Ixx = 0.59,Iyy = 0.58, Izz = 1.15, and Iij ≈ 0,∀i 6= j.

Figure 8: The NASA Flying Test Platform.

4.2 Bench Testing

In order to generate control design models and performdetail dynamical simulations which are physically rele-vant, we first characterize by parameterizing the dynam-ics of the subsystems and total system from first princi-

ples and then use a series of bench tests to determine theseparameters from measured data. Specific necessary tasksto this end includes design and fabrication of specific testconfigurations, instrumentation, calibration, followed bymodel development and validation through estimation ofparameters including noise levels and its associated fil-ters.

Figure 9 shows the bench test configuration for anindividual ducted fan and vane system. The focus in thisbench test involves characterizing (i) Fan motor and vanedynamics, and (ii) Fan and vane aerodynamics. The for-mer basically involves the development of models to char-acterize fan speed and vane angle from their commands,while the latter modeling task can be anywhere from al-most trivial to near impossible if the unsteady aero effectsare to be included with any level of confidence. Togetherthey determine the actual net forces and torques gener-ated by control commands for fan speeds and vane deflec-tion angles.

Ob

WiT

−TLC

−τLC

τLC

TLC

rLC

gm x

Wi

D1

L 1M1

M2 L 2

D2

a i^

Load Cell

Bench test stand

Oi

F

biF

x i

τ

z i

F

F

Figure 9: Bench test configurations and modeling.

Initially we consider a static aero model for the con-trol forces and torques generated from fan speed and vaneangle. For example, consider determining a steady aeromodel of net forces due to fan and vane. Figure 10 showsthe measured net forces as a function of fan speed, for afixed vane angle, while Figure 11 shows normalized mea-sured vertical forces as a function of van angle, at constantfan speeds. Figure 12 illustrates the measured unsteadyeffects in the vertical force component with time vary-ing fan speeds with fixed vane angle. The 1-σ variationof 0.1 pound thrust in a single fan is not insignificantfor this subscale platform, especially considering that wehave not included additional factors such as vane effectsand implicit bandwidth limitations in the fan motor.

The above sample data were obtained experimentally,by testing the components in a ducted fan configuration,displayed in Figure 9. Static thrust vector force compo-nents were obtained from these tests, and are provided in[8]. Unsteady flow forces which are difficult to character-ize are not quantified, and do not appear in the simulationmodel described in the next Section.

9

Figure 10: Measured forces with fan speeds, at fixed 0degree vane angle.

4.3 Simulation

In this section we present simulation results based on fullnonlinear models that are constructed from first princi-ples, bench test data refined model parameters, sampledand quantized measurements, and saturation limits onactuators. In particular, the dynamical models for theDucted fan/vane, whose development are based on benchtests, includes asymmetrical effects which are difficult tovisualize due to their 3-dimensional characteristics. ThisMATLAB- Simulink simulation model also incorporatesmodels of servo dynamics that regulate fan speeds andvane angles.

In all simulation cases which follows, the vanes andthe fan speeds are actively used for stabilization and con-trol which reflects the current state-of-the art approachwhich is vector and differential thrusting, albeit, RC-quality indoor lab experiments. Initially, we focus oursimulation on the effects of bias momentum in the stabi-lization and control performance of our flying test plat-form and do not include CMGs. We highlight these effectsby comparing a system with significant level of bias mo-mentum (BM cases) with a corresponding system withno bias momentum (NMB). A difference in our simula-tion studies is to independently control all four fan speedsand their respective vane angles, as opposed to constrain-ing it to work in pairs and collectively, as in conventionalhelicopter control [18],[19] or practically all VTOL flyingplatforms [1].

4.3.1 Crosswind and turbulence model

To simulate wind disturbances during hovering or low air-speed operations, crosswinds are assumed which impingeson a vertical cylinder attached to the upper part of theplatform. The resulting aerodynamic drag on this cylin-

Figure 11: Measured and quadratic fit vertical force withvane angles, at different fan speeds.

drical column is used to simulate the crosswind and tur-bulence on the vehicle. The resulting disturbance modelconsists of forces and torques in the pitch and roll axis.Two sets of winds are simulated, namely, “light” winds,and “strong” winds. The “light” winds simulate “Calmto Gentle Breeze” with sample mean wind speeds of 7Knots and standard deviation of 5. However for “strong”winds, intended to simulate “Calm to Strong Breeze/NearGale”, the simulated sample mean wind speed is 15 Knotswith a standard deviation of 18, as seen in Figure 13.

4.3.2 Controller Law Synthesis

Based on LQG control theory, feedback controllers aredesigned to stabilize and hold the platform attitude andalso track the translational velocity and angular velocitycommands. In this simulated design study, feedback con-trollers for the BM and NBM cases are synthesized basedon the linearized model about hovering trim, which isidentical to the trim conditions at a given translationalvelocity if aerodynamics due to significant airspeeds arenot included, as assumed in this study. Note that thetrim conditions are necessarily different for both casessince aero-resistance fan drag torques in the case must bestatically canceled by vane-deflected forces.

The augmented linearized NFTP model with fan andservo models has 21 states consisting of 3 attitude pa-rameters, 3 linear momentum, 3 angular momentum, 8fan motor states, and 4 servo motor states. The 8 fanmotor states includes the PI controllers which regulateeach fan speed. The outputs of the augmented modelare translational and angular velocities, both expressedin vehicle fixed coordinates. For a fair comparison be-tween BM and NBM cases, the same LQG weights wereused in the controller design

10

Figure 12: Measured uncertainty in the vertical force withtime varying fan speeds at a fixed vane angle.

Figure 14 shows the tradeoff between performancecost and control effort for a parameterized set of LQGcontrollers undergoing gust response during hover over aperiod of time Tfinal. Specifically, the performance costin terms of attitude and position errors are

J(t) ,∫ t

0

3∑i=1:3

[(βi − βi)2 + (ξi − ξi)2]dt (46)

and the control effort is defined by a cumulative controleffort (CCA) function as follows

CCA(t) ,∫ t

0

(θ21 + θ22 + θ23 + θ24)dt (47)

The abscissa and ordinate in Figure 14 representsCCA(Tfinal) and J(Tfinal) respectively. In addition toquantifying the level of control activity, the above con-trol effort is intended to capture the level of control vaneactivity, which is seen as the primary source of controlsurface induced unsteady aerodynamics due to its vectorthrusting role. Albeit limited, this tradeoff study indi-cates that controllers are significantly more effective for asystem endowed with significant levels of bias momentum(asterisk line) in terms of actuator effort necessary for agiven level of performance cost. The figure also shows theeffects of a hypothetical model of unsteady aero depen-dence on actuator effort [8], namely, the expectation thatcontrolled performance can inadvertently degrade withoverly active control surface activity as mostly seen inthe NBM case (diamond line). However, in the followingsimulations, this hypothetical model of controller inducedunsteady aerodynamics will not be included.

0 5 10 15 20−10

0

10

20

X−

forc

e (N

)

0 5 10 15 20−20

−10

0

10

20

Y−

forc

e (N

)

0 5 10 15 20−4

−2

0

2

4

X−

torq

ue (

N−

m)

0 5 10 15 20−2

0

2

4

6

Y−

torq

ue (

N−

m)

0 5 10 15 20−50

0

50

Win

d X

−ve

l. (k

nots

)

Time(sec)0 5 10 15 20

−50

0

50

win

d Y

−ve

l. (k

nots

)

Time(sec)

Figure 13: Simulated strong wind.

4.3.3 Hovering

Open Loop Response to Light Winds. The open-loop attitude responses of the initially trimmed system forthe two cases are shown in Figure 15. In the NBM case,the vehicle rapidly loses orientation and goes unstable,but in the BM case, the vehicle holds attitude inspite ofa small gyric roll and pitch angle response to the lightwinds. However, the platform drifts off slowly withoutfeedback compensation due to a steady wind componentin the light winds.

Closed Loop Response to Light Winds. Theclosed-loop responses to light winds during hover for bothcases are shown in Figure 16. Although both cases showstable response, the angular response for the BM caseis clearly superior to the NBM case. Apparently, un-der these simulated light wind conditions, the angular re-sponse amplitudes for the NBM case appears to be smallenough to be tolerated during an actual flight. The timehistory of the cumulative control effort (bottom right ofFigure 16), shows that the case with BM actually requiresless vane actuation than in the corresponding NBM case.As expected, the vane and fan actuator response did notsaturate for both cases, under light winds.

Closed Loop Response to Strong Winds. Theclosed-loop responses to strong winds during hover forboth cases are shown in Figure 17. The controller forNBM case was judiciously scaled to mitigate actuatorsaturation. Both cases remains stable but the angularresponse for the BM case is clearly superior to the NBMcase. However, the significantly larger angular responseamplitudes for the NBM case will likely be unacceptableor even dangerous during an actual flight. The time his-tory of the cumulative control effort shows that the case

11

Figure 14: Performance cost (ordinate) vs actuator effort(abscissa) for an LQG set of controllers undergoing gustresponse during hover.

with BM actually requires less vane actuation althoughit gives far superior attitude hold than the correspondingNBM case. Some saturation in the vane and fan actuatorresulted for the NBM case.

To examine what happens if the controller for NBMcase is not judiciously scaled to mitigate actuator sat-uration, resulting possibly to a more agressive feedbackcontrol, an additional simulation was done with the newcontrol law. It turns out that the close loop system losescontrol after a few seconds. More specifically Figure 18shows the time histories of the vane angle and fan speedresponses. The vane and fan actuators saturate resultingin loss of vehicle under strong winds.

4.3.4 Command Tracking

In this section, we outline a sample of simulation results ofcommand tracking under strong winds, specifically trans-lational velocity commands while holding a trim attitude.It should be noted that the demonstrated inherent advan-tage of bias momentum in endowing directional stabilityto the platform do not necessarily improve its maneuver-ability. Hence, the simulation results in this subsection isintended to examine this issue quantitatively.

For the following simulations, LQG based controllersare designed to track a trajectory path defined by a com-manded translational velocities about trim with zero an-gular velocity. The input commands are shaped to mit-igate the discontinuity in a step command. Figure 19shows the response to translational velocity tracking com-mands for both cases subject to strong winds. The accom-panying excursions in the angular velocities (and hencethe vehicle attitude) are shown in Figure 20. It is seenthat for both cases, the vehicle can successfully tracktranslational velocity commands but BM case gives su-

0 5 10 15 20 25

−150

−100

−50

0

50

100

150

Rol

l (de

g)

bmnbm

0 5 10 15 20 25

−50

0

50

Pitc

h (d

eg)

0 5 10 15 20 25

−150

−100

−50

0

50

100

150

Yaw

(de

g)

Time(sec)0 5 10 15 20 25

0

0.05

0.1

0.15

0.2

CC

A

Time(sec)

Figure 15: Open loop response to light winds, 3-2-1 Eulerangles.

perior performance.The corresponding vane and fan actuator time histo-

ries during tracking are shown in Figures 21 and 22, re-spectively. The case with BM clearly needs less vane andfan control activity but as observed earlier gives smallervelocity tracking errors.

4.3.5 Robustness to payload variations

Simulated Mass & Inertia changes. To simulatevariable payloads, we assume a step increase of 20plat-form. A spherical uniformly distributed mass is assumeddropped at (x=0.15,y=0,z=0) location on the platform at5 sec and is assumed to be dropped off from the platformat 20 sec. The inertia matrix is significantly changed asa result of the off-centered loading. This off-CG centeredloading simulation is of particular interest due to its diffi-culty and its practical significance for a hovering platformwhere passengers or payloads can be loaded and unloadedwith maximum flexibility without compromising vehiclestability while hovering. In this part of simulation, dis-turbances and measurement errors are not included tofocus entirely on the effects of these payload changes.

Closed loop response during hover. Figures 23 and24 show the position and attitude responses to mass andinertia changes for both cases. It is seen that the platformposition and attitude excursions are significantly betterin the BM case.

Significant differences are also noted in their corre-sponding vane and fan actuator histories in Figure 25. Itis seen that the fan speeds and vane angles oscillates sig-nificantly and rapidly in the NBM case resulting in theplatform oscillations shown previously. Unfortunately,the actual consequences on flight performance due to this

12

0 5 10 15 20 25−30

−20

−10

0

10

20

30

Van

e 1 (de

g)

bmnbm

0 5 10 15 20 25−30

−20

−10

0

10

20

30

Van

e 2 (de

g)

0 5 10 15 20 25−30

−20

−10

0

10

20

30

Van

e 3 (de

g)

Time(sec)0 5 10 15 20 25

−30

−20

−10

0

10

20

30

Van

e 4 (de

g)

Time(sec)

0 5 10 15 20 250

1

2

3

4x 10

4

ws 1 (

rpm

)

bmnbm

0 5 10 15 20 250

1

2

3

4x 10

4

ws 2 (

rpm

)

0 5 10 15 20 250

1

2

3

4x 10

4

ws 3 (

rpm

)

Time(sec)0 5 10 15 20 25

0

1

2

3

4x 10

4

ws 4 (

rpm

)

Time(sec)

Figure 18: Closed loop response to strong winds, more agressive control.

0 5 10 15 20 25−2

−1

0

1

2

3

Rol

l (de

g)

bmnbm

0 5 10 15 20 25−2

−1.5

−1

−0.5

0

0.5

1

1.5

Pitc

h (d

eg)

0 5 10 15 20 25−3

−2

−1

0

1

2

3

Yaw

(de

g)

Time(sec)0 5 10 15 20 25

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

CC

A

Time(sec)

Figure 16: Closed loop response to light winds.

apparently more unsteady control vane activity for theNBM case, remains to be seen during an actual flight dueto the lack of a reliable mathematical model for this classof unsteady aero.

5 Conclusions

A new stability and control technology, based on trying tomitigate control surface actuator induced unsteady aeroeffects in feedback control is proposed. Its ultimate goal isto enable statically unstable thrust levitated low-speed orhovering vehicles to operate more safely than is possibletoday, in turbulent conditions and under large payloadvariations. New results in trim, bias momentum sizing,

0 5 10 15 20 25−30

−20

−10

0

10

20R

oll (

deg)

bmnbm

0 5 10 15 20 25−30

−20

−10

0

10

20

Pitc

h (d

eg)

0 5 10 15 20 25−80

−60

−40

−20

0

20

40

Yaw

(de

g)

Time(sec)0 5 10 15 20 25

−100

0

100

200

300

400

500

CC

A

Time(sec)

Figure 17: Closed loop response to strong winds.

and a control structure framework which integrates con-ventional vector thrusting with multiple wheel actuatorshave been developed. Simulation results indicate thatgeneric VTOL platforms can be endowed with signifi-cant directional stability with a proper amount of biasmomentum, similar to a dual-spin stabilized spacecraftbut under a significantly different operating environment.The results also indicate that this open loop directionalstability carries over to closed loop if the controllers aredesigned properly. Specifically, it was found that vehicleattitude robustness can be significantly improved (overa no-bias-momentum vehicle) during hovering, particu-larly under strong turbulent winds or significant payloadvariations. In addition, command tracking performanceat low airspeeds can also be improved significantly par-

13

0 5 10 15 20 25−15

−10

−5

0

5

10

Van

e 1 (de

g)

bmnbm

0 5 10 15 20 25−15

−10

−5

0

5

10

Van

e 2 (de

g)

0 5 10 15 20 25−10

−5

0

5

10

15

20

Van

e 3 (de

g)

Time(sec)0 5 10 15 20 25

−15

−10

−5

0

5

10

15

20

Van

e 4 (de

g)

Time(sec)

0 5 10 15 20 252.6

2.8

3

3.2

3.4

3.6x 10

4

ws 1 (

rpm

)

bmnbm

0 5 10 15 20 252.7

2.8

2.9

3

3.1

3.2

3.3x 10

4

ws 2 (

rpm

)

0 5 10 15 20 252.6

2.7

2.8

2.9

3

3.1

3.2x 10

4

ws 3 (

rpm

)

Time(sec)0 5 10 15 20 25

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3x 10

4

ws 4 (

rpm

)

Time(sec)

Figure 25: Actuator responses to step changes in payload.

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

Vx (m

/s)

bmnbmcmd

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

Vy (m

/s)

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

Vz (m

/s)

Time(sec)

Figure 19: Vehicle velocity response during trajectorytracking.

ticularly under turbulent winds, despite the presence ofsomewhat complicated gyroscopic coupling. These per-formance improvements are due to control laws that arebased on proper modeling of the system dynamics. Thesepredicted improvements in robustness obtained from de-tailed simulation studies are expected to be more evidentduring actual flight test, with the physical presence ofcontrol induced unsteady aerodynamics.

AcknowledgementsThe authors would like to thank their colleagues Elvin

Ahl for developing and providing the CAD figure of on-going NFTP test configuration, and Plesent Goode for

0 5 10 15 20 25 30 35 40 45 50

−2

0

2w

x (rad

/s)

bmnbm

0 5 10 15 20 25 30 35 40 45 50

−2

0

2

wy (r

ad/s

)

0 5 10 15 20 25 30 35 40 45 50

−2

0

2

wz (r

ad/s

)

Time(sec)

Figure 20: Vehicle attitude response during trajectorytracking.

giving tangible moral support for this research.

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14

0 10 20 30 40 50−30

−20

−10

0

10

20

30

Van

e 1 (de

g)

bmnbm

0 10 20 30 40 50−30

−20

−10

0

10

20

30

Van

e 2 (de

g)

0 10 20 30 40 50−30

−20

−10

0

10

20

30

Van

e 3 (de

g)

Time(sec)0 10 20 30 40 50

−30

−20

−10

0

10

20

30

Van

e 4 (de

g)

Time(sec)

Figure 21: Vane angle response during tracking.

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0 10 20 30 40 500

1

2

3

4x 10

4

ws 1 (

rpm

)

bmnbm

0 10 20 30 40 500

1

2

3

4x 10

4

ws 2 (

rpm

)

0 10 20 30 40 500

1

2

3

4x 10

4

ws 3 (

rpm

)

Time(sec)0 10 20 30 40 50

0

1

2

3

4x 10

4

ws 4 (

rpm

)

Time(sec)

Figure 22: Fan speed response during command tracking.

0 5 10 15 20 25−1

0

1

2

X (m

)bmnbm

0 5 10 15 20 25−0.5

0

0.5

Y (m

)

0 5 10 15 20 25−1.5

−1

−0.5

0

0.5

Z (m

)

Time(sec)

Figure 23: Position response to step changes in payload.

Balas, Tutorial Workshop at the American ControlConference, June 2-3, 1997, Albuquerque, NM.

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15

0 5 10 15 20 25−10

−5

0

5

10

Rol

l (de

g)

bmnbm

0 5 10 15 20 25−15

−10

−5

0

5

10

Pitc

h (d

eg)

0 5 10 15 20 25−20

−10

0

10

20

30

40

Yaw

(deg

)

Time(sec)

Figure 24: Attitude response to step changes in payload.

[16] Tsiotras, P., “Stabilization and Optimality Resultsfor the Attitude Control Problem,” Journal of Guid-ance, Control, and Dynamics, Vol. 19, No. 4, 1996,pp. 772-9.

[17] Schaub,H., Vadali,S.R., and Junkins,J.L., “Feedbackcontrol law for variable speed control moment gy-ros,” Journal of the Astronautical Sciences, Vol. 46,No. 3, July-Sept. 1998, pp. 307-328.

[18] George H. Saunders, Dynamics of Helicopter Flight,John Wiley & Sons, Inc, New York, 1974.

[19] Prouty, R.W., Helicopter Aerodynamics, Vols. I-III,PJS Publications Inc, Peoria, Ill, 1985-1993.

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